Similar Figures and Indirect Measurement
Use scale factors and proportional reasoning to find missing dimensions in similar figures — and measure heights and distances you can't reach directly.
Indirect measurement lets you find the height of a tree, a building, or a mountain without climbing it — just by comparing shadows or using similar triangles. Architects build scale models. Map makers use scale factors to represent the real world. Photographers understand how image sensors create similar triangles with their subjects. Any time you can't measure something directly, similar figures give you the mathematical tool to figure it out.
- Define similar figures and identify corresponding sides and angles
- Calculate and apply scale factors to find missing side lengths
- Use proportions to solve indirect measurement problems (shadows, mirrors, maps)
If triangles ABC and DEF are similar (\(\triangle ABC \sim \triangle DEF\)), then:
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = \text{scale factor} \]All corresponding sides share the same ratio. Cross-multiply any two ratios to find a missing side — this is the proportion-solving technique from ratios applied to geometry.
Shadow Method — Indirect Measurement
A 6-foot person casts a 4-foot shadow at the same time a tree casts an 18-foot shadow.
The person and their shadow form a triangle similar to the tree and its shadow:
\[ \frac{\text{person height}}{\text{person shadow}} = \frac{\text{tree height}}{\text{tree shadow}} \] \[ \frac{6}{4} = \frac{h}{18} \Rightarrow h = \frac{6 \times 18}{4} = 27 \text{ feet} \]The tree is 27 feet tall — measured without a ladder or measuring tape.
Two similar rectangles: the first is 8 cm × 5 cm. The second has a length of 20 cm. Find its width.
- Scale factor: \(\frac{20}{8} = 2.5\)
- Width: \(5 \times 2.5 = 12.5\) cm
The second rectangle is 20 cm × 12.5 cm.
Triangle PQR ~ Triangle XYZ. PQ = 12, QR = 15, PR = 9. XY = 8. Find YZ and XZ.
- Scale factor: \(\frac{XY}{PQ} = \frac{8}{12} = \frac{2}{3}\)
- YZ: \(15 \times \frac{2}{3} = 10\)
- XZ: \(9 \times \frac{2}{3} = 6\)
The smaller triangle has sides 8, 10, 6.
A map has a scale of 1 inch = 25 miles. Two cities are 3.6 inches apart on the map. What is the actual distance?
\[ \frac{1 \text{ in}}{25 \text{ mi}} = \frac{3.6 \text{ in}}{d} \Rightarrow d = 3.6 \times 25 = 90 \text{ miles} \]The cities are 90 miles apart in reality.
- Two similar triangles have corresponding sides 5 and 15. What is the scale factor?
- A 5-ft post casts a 3-ft shadow. A nearby flagpole casts a 21-ft shadow. How tall is the flagpole?
- On a blueprint with scale 1 cm = 4 m, a room measures 3.5 cm × 2.5 cm. What are the actual dimensions?
▶ Show Answers
- Scale factor = 3 (15 ÷ 5).
- \(\frac{5}{3} = \frac{h}{21}\) → \(h = 35\) ft. The flagpole is 35 feet tall.
- \(3.5 \times 4 = 14\) m by \(2.5 \times 4 =\) 10 m. The room is 14 m × 10 m.
- Matching the wrong sides: Always match corresponding sides (same position in both figures). In similar triangles, match the sides opposite the same angle.
- Confusing similar and congruent: Congruent = same shape AND same size (scale factor = 1). Similar = same shape, different size.
- Forgetting to use the same units: In the shadow problem, if the person's height is in feet, the shadow must also be in feet. Mixed units break the proportion.
- Similar figures: equal corresponding angles, proportional corresponding sides.
- Scale factor = ratio of any pair of corresponding sides — apply to ALL sides.
- Set up proportions with corresponding sides and cross-multiply to find missing values.
- Indirect measurement uses similar triangles formed by objects, shadows, or mirrors.
Architects work with scale drawings where every measurement on paper corresponds to a real dimension via a scale factor. Cartographers (mapmakers) design scales so users can calculate real distances from map measurements. Photographers and opticians work with similar triangle relationships between object size, distance, and image size. Every scaled model, blueprint, and map in existence is built on similar figure mathematics.
Calculator Connection
The Pythagorean Theorem Calculator can verify side relationships in similar right triangles. The Triangle Angle Calculator confirms angle equality needed for AA similarity.
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Similar Figures and Indirect Measurement — Quiz
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